With the subscript on the bias function, we follow [1]} by making explicit that this function depends on \(\theta \) . [2]} imposed a zero mean Gaussian process prior on the unknown bias function, i.e. \(\delta _{\theta } (\cdot ) \sim \mathrm {GP}\left( 0, \tau ^2 \kappa (\cdot , \cdot ; \rho ) \right)\)  where \(\kappa (\cdot , \cdot ; \rho )\)  is a specified correlation function depending on an \(r \times 1\)  vector of parameters \(\rho \) . A fully Bayesian approach was taken by evaluating the marginal posterior distribution of \(\theta \)  after specifying prior distributions for \(\theta \) , \(\sigma ^2\) , \(\rho \)  and \(\tau ^2\) .
